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Fix projective space bundle formula for motives
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aisejohan committed Jul 19, 2021
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8 changes: 4 additions & 4 deletions weil.tex
Expand Up @@ -1167,14 +1167,14 @@ \section{Projective space bundle formula}
c_i = c_1(\text{pr}_2^*\mathcal{O}_P(1))^i \cap [\Gamma_p]
\in \text{Corr}^i(X, P)
$$
We may and do think of $c_i$ as a morphism $h(X)(i) \to h(P)$.
We may and do think of $c_i$ as a morphism $h(X)(-i) \to h(P)$.

\begin{lemma}[Projective space bundle formula]
\label{lemma-projective-space-bundle-formula}
In the situation above, the map
$$
\sum\nolimits_{i = 0, \ldots, r - 1} c_i :
\bigoplus\nolimits_{i = 0, \ldots, r - 1} h(X)(i)
\bigoplus\nolimits_{i = 0, \ldots, r - 1} h(X)(-i)
\longrightarrow
h(P)
$$
Expand Down Expand Up @@ -1232,11 +1232,11 @@ \section{Projective space bundle formula}
for all $i \in \{0, \ldots, r - 1\}$. Thus we see that the matrix
of the composition
$$
\bigoplus h(X)(i)
\bigoplus h(X)(-i)
\xrightarrow{\bigoplus c_i}
h(P)
\xrightarrow{\bigoplus c'_j}
\bigoplus h(X)(j)
\bigoplus h(X)(-j)
$$
is invertible (upper triangular with $1$s on the diagonal).
We conclude from the projective space bundle formula
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